In geometry, the equation of a line can be written in different forms and each of these representations is useful in different ways. The equation of a straight line is written in either of the following methods:

- Point slope form
- Two point form
**Slope intercept form**- Intercept form

In this article, you will learn about one of the most common forms of the equation of lines called slope-intercept form along with derivation, graph and examples.

Learn what is the intercept of a line here.

Let’s have a look at the slope-intercept form definition.

## What is the Slope Intercept Form of a Line?

The graph of the linear equation *y* = *mx* + c is a line with m as slope, *m* and c as the y-intercept. This form of the linear equation is called **the slope-intercept form**, and the values of m and c are real numbers.

The **slope, **** m**, represents the steepness of a line. The slope of the line is also termed as gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis.

**Also, read:**

- Straight Lines Class 11
- Linear Equation In One Variable
- Slope of a line

## Slope Intercept Form Equation

In this section, you will learn the derivation of the equation of a line in the slope-intercept form.

Consider a line L with slope m cuts the y-axis at a distance of c units from the origin.

Here, the distance c is called the y-intercept of the given line L.

So, the coordinate of a point where the line L meets the y-axis will be (0, c).

That means, line L passes through a fixed point (0, c) with slope m.

We know that, the equation of a line in point slope form, where (x_{1}, y_{1}) is the point and slope m is:

(y – y_{1}) = m(x – x_{1})

Here, (x_{1}, y_{1}) = (0, c)

Substituting these values, we get;

y – c = m(x – 0)

y – c = mx

y = mx + c

Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if** y = mx + c**

**Note:** The value of c can be positive or negative based on the intercept is made on the positive or negative side of the y-axis, respectively.

### Slope Intercept Form Formula

As derived above, the equation of the line in slope-intercept form is given by:

y = mx + c

Here,

(x, y) = Every point on the line

m = Slope of the line

c = y-intercept of the line

Usually, x and y have to be kept as the variables while using the above formula.

### Slope Intercept Form x Intercept

We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:

y = m(x – d)

Here,

m = Slope of the line

d = x-intercept of the line

Sometimes, the slope of a line may be expressed in terms of tangent angle such as:

m = tan θ

**Also, try: **Slope Intercept Form Calculator

### Derivation of Slope-Intercept Form from Standard Form Equation

We can derive the slope-intercept form of the line equation from the equation of a straight line in the standard form as given below:

As we know, the standard form of the equation of a straight line is:

Ax + By + C = 0

Rearranging the terms as:

By = -Ax – C

⇒y = (-A/B)x + (-C/B)

This is of the form y = mx + c

Here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.

## Slope Intercept Form Graph

When we plot the graph for slope-intercept form equation we get a straight line. Slope-intercept is the best form. Since it is in the form “y=”, hence it is easy to graph it or solve word problems based on it. We just have to put the x-values and the equation is solved for y.

The best part of the slope-intercept form is that we can get the value of slope and the intercept directly from the equation.

## Solved Examples

**Example 1: **

Find the equation of the straight line that has slope m = 3 and passes through the point (–2, –5).

**Solution**:

By the slope-intercept form we know;

y = mx+c

Given,

m = 3

As per the given point, we have;

y = -5 and x = -2

Hence, putting the values in the above equation, we get;

-5 = 3(-2) + c

-5 = -6+c

c = -5 + 6 = 1

Hence, the required equation will be;

y = 3x+1

**Example 2: **

Find the equation of the straight line that has slope m = -1 and passes through the point (2, -3).

**Solution:**

By the slope-intercept form we know;

y = mx+c

Given,

m = -1

As per the given point, we have;

y = -3 and x = 2

Hence, putting the values in the above equation, we get;

-3 = -1(2) + c

-3 = -2 + c

c = -3+2 = -1

Hence, the required equation will be;

y = -x-1

**Example 3:**

Find the equation of the lines for which tan θ = 1/2, where θ is the inclination of the line such that:

(i) y-intercept is -5

(ii) x-intercept is 7/3

**Solution:**

Given, tan θ = 1/2

So, slope = m = tan θ = 1/2

(i) y-intercept = c = -5

Equation of the line using slope intercept form is:

y = mx + c

y = (1/2)x + (-5)

Or

2y = x – 10

x – 2y – 10 = 0

(ii) x-intercept = d = 7/3

Equation of slope intercept form with x-intercept is:

y = m(x – d)

y = (1/2)[x – (7/3)]

Or

2y = (3x – 7)/3

6y = 3x – 7

3x – 6y – 7 = 0

## Practice Problems

- Find the slope of the line y = 5x + 2.
- Find the slope of the line which crosses the line at point (-2,6) and have an intercept of 3.
- What is the equation of the line whose angle of inclination is 45 degrees and x-intercept is -⅗?
- Write the equation of the line passing through the point (0, 0) with slope -4.